The concept of consistency of interaction between the gauge fields
has been developed in [1]}. The consistent couplings between
gauge vectors has been studied in [2]}, [3]}. It has been
shown that the Yang-Mills vertex is unique in the covariant setting.
The Lagrangian self-interactions of the gauge vector multiplet
subjected to the ECS equations at free level has been reconsidered
in article [4]}. The most general consistent non-linear
theory is proven to have the Yang-Mills gauge symmetry, and the
Lagrangian is given by the covariantization of the free ECS
Lagrangian. The interacting model demonstrates the Ostrogradski
instability at the non-linear level in all the instances, even
though the free theory is stable. This result implies a no-go
theorem for stable Lagrangian interactions in the ECS model. The
conclusion is not surprising because the Lagrangian couplings
preserve the conservation law of canonical energy, being unbounded
already at the free level. The inclusion of non-Lagrangian
interactions can solve the issue of the dynamics stability at the
interacting level, because such couplings preserve bounded conserved
quantities. However, the problem of construction of stable
consistent non-Lagrangian (self-)couplings (stable or unstable) has
not been studied in the ECS model in the literature before.
